2) Prove or disprove the following statements: (a) “If A € M5(R) has a non-real eigenvalue,...
2) Prove or disprove the following statements: (a) "If A E M5(R) has a non-real eigenvalue, then A is diagonalizable." (b) "If 2 E C" is an eigenvector of A € Mn(C) then Z is also an eigenvector of A."
Assume all matricies are Mmxm(R) unless otherwise specified. 1. (1 point) Prove or disprove that the eigenvalues of A and AT are the same. 2. (2 points) Let A be a matrix with m distinct, non-zero, eigenvalues. Prove that the eigenvectors of A are linearly independent and span R”. Note that this means in this case) that the eigenvectors are distinct and form a base of the space. 3. (1 point) Given that is an eigenvalue of A associated with...
4. Assume that A, B E Mnxn(R). Prove or disprove each of the following statements. (a) If AB is a product of elementary matrices, then A is a product of elementary matrices. (b) If R is the RREF of A, then det A = det R. (c) If det A-det B, then A = B.
Question 4: Eigenvalue Theory 2 Let A Cnxn. For each of the following statements show that it is true or give a counterexample to show that it is false (a) If λ is an eigenvalue of A, and μ є Cn then λ-μ is an eigenvalue of A-1 (b) If A is real and λ is an eigenvalue of A then so is-λ. (c) If A is real and λ is an eigenvalue of A, then so is λ. (d)...
Prove or disprove the following. (a) R is a field. (b) There is an additive identity for vectors in R^n. (If true, what is it?)........ 1. Prove or disprove the following. (a) R is a field (b) There is an it?) additive identity for vectors in R". (If true, what is (c) There is a is it? multiplicative identity for vectors in R". (If true, what (d) For , , (e) For a, bE R and E R", a(b) =...
Please do number 2 Assume all matricies are Mmxm(R) unless otherwise specified. 1. (1 point) Prove or disprove that the eigenvalues of A and AT are the same. 2. (2 points) Let A be a matrix with m distinct, non-zero, eigenvalues. Prove that the eigenvectors of A are linearly independent and span R”. Note that this means in this case) that the eigenvectors are distinct and form a base of the space. 3. (1 point) Given that is an eigenvalue...
linear algebra question 2. (5' each) Give short answers: (a) True or false: If Ai-Adi for some real number λ, then u is an eigenvector of matrix A. If a square matrix is diagonalizable, then it has n distinct real eigenvalues. Two vectors of the same dimension are linearly independent if and only if one is not a multiple of the other. If the span of a set of vectors is R", then that set is a basis of R...
Let v 2 Rn be a unit vector. Define G = I ? vvT . (a) Show G is symmetric and G2 = G. (b) Prove v is an eigenvector, find the associated eigenvalue. (c) Prove that if < u; v >= 0 then u is also an eigenvector of G. (d) Prove that G is diagonalizable. Let v ER" be a unit vector. Define G=I - vt. (a) Show G is symmetric and G =G. (b) Prove v is...
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B = C (b) If Bvi,.., Bvh} is a then vi, . ., vk} is a linearly independent set in R". linearly independent set in R* where B is a kx n matrix, 5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B...
True False a) For nxn A, A and AT can have different eigenvalues. b) The vector v 0 cannot be an eigenvector of A. c) If λ's an eigenvalue of A, then λ2 is an eigenvalue of A2. True False d) If A is invertible, then A is diagonalizable. e) If nxn A is singular, then Null(A) is an eigenspace of A. f) For nxn A, the product of the eigenvalues is the trace of A. True False g) If...